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Squeeze operator

The squeeze operator, denoted as \hat{S}(\zeta), is a unitary operator in quantum mechanics and quantum optics, mathematically expressed as \hat{S}(\zeta) = \exp\left[ \frac{1}{2} (\zeta^* \hat{a}^2 - \zeta \hat{a}^{\dagger 2}) \right], where \hat{a} and \hat{a}^\dagger are the annihilation and creation operators for a bosonic mode, and \zeta = r e^{i\theta} is a complex parameter with squeeze factor r \geq 0 and phase \theta.^[1]^[2] Applying this operator to the vacuum state |0\rangle produces a squeezed vacuum state, which exhibits reduced quantum fluctuations (noise) in one quadrature of the electromagnetic field—such as the amplitude or phase—below the standard quantum limit of coherent states, while the uncertainty in the orthogonal quadrature increases to satisfy the Heisenberg uncertainty principle.^[3]^[4] Introduced theoretically in the early 1980s as part of the broader framework of nonclassical light states, the squeeze operator builds on the displacement operator \hat{D}(\alpha) to form squeezed coherent states |\alpha, \zeta\rangle = \hat{D}(\alpha) \hat{S}(\zeta) |0\rangle, enabling phase-sensitive noise reduction without classical analogs.^[3]^[4] Key properties include its unitarity (\hat{S}^\dagger(\zeta) = \hat{S}(-\zeta)), which preserves normalization, and its transformation of operators: \hat{S}^\dagger(\zeta) \hat{a} \hat{S}(\zeta) = \hat{a} \cosh r - \hat{a}^\dagger e^{i\theta} \sinh r, leading to quadrature variances \langle (\Delta \hat{X}_\theta)^2 \rangle = \frac{1}{4} e^{\pm 2r} for appropriately chosen angles.^[1]^[2] These states are generated experimentally via nonlinear optical processes, such as parametric down-conversion in crystals or four-wave mixing in atomic vapors, achieving squeezing levels up to 15 dB in laboratory settings.^[5] In applications, squeezed states facilitated by the squeeze operator enhance precision in interferometric measurements, notably improving the sensitivity of gravitational wave detectors like LIGO by reducing shot-noise limits.^[5] They also play a crucial role in quantum information science, including continuous-variable quantum computing, secure communication protocols, and entanglement generation for multipartite systems, where two-mode squeeze operators \hat{S}_2(\zeta) = \exp\left[ \zeta^* \hat{a}_1 \hat{a}_2 - \zeta \hat{a}_1^\dagger \hat{a}_2^\dagger \right] create correlated photon pairs.^[1]^[5] Despite challenges from optical losses that degrade squeezing, ongoing advances in integrated photonics and optomechanics continue to expand their utility in surpassing classical performance bounds.^[6]

Introduction

Definition

The squeeze operator in quantum optics is a unitary transformation defined by

\hat{S}(z) = \exp\left[\frac{1}{2}\left(z^* \hat{a}^2 - z (\hat{a}^\dagger)^2\right)\right],

where z = r e^{i\theta} is a complex parameter, with r \geq 0 denoting the squeeze parameter that controls the degree of squeezing and \theta the squeeze phase that determines the orientation of the squeezed quadrature, and \hat{a}, \hat{a}^\dagger the annihilation and creation operators of the bosonic mode, respectively. This operator is unitary, satisfying \hat{S}^\dagger(z) \hat{S}(z) = \hat{S}(z) \hat{S}^\dagger(z) = \hat{1}, which ensures it maps states within the Hilbert space to other valid quantum states while preserving inner products and probabilities. Conceptually, the squeeze operator generates states with reduced quantum noise in one field quadrature—analogous to position or momentum—below the standard vacuum fluctuation level, at the expense of increased noise in the orthogonal quadrature, consistent with the Heisenberg uncertainty principle \Delta X \Delta P \geq 1/4. Squeezed states result from applying \hat{S}(z) to the vacuum or coherent states.

Physical motivation

In quantum optics, the behavior of light is described using quadrature operators that represent the amplitude and phase components of the electromagnetic field for a single mode. These are defined as \hat{X} = \frac{1}{2}(\hat{a} + \hat{a}^\dagger) and \hat{P} = \frac{1}{2i}(\hat{a} - \hat{a}^\dagger), where \hat{a} and \hat{a}^\dagger are the annihilation and creation operators satisfying [\hat{a}, \hat{a}^\dagger] = 1. The quadratures obey the commutation relation [\hat{X}, \hat{P}] = \frac{i}{2}, which implies the Heisenberg uncertainty principle \Delta X \Delta P \geq \frac{1}{4}. For the vacuum state, the uncertainties are symmetric, with \Delta X = \Delta P = \frac{1}{2}, setting a baseline quantum noise level.^[3] Classical treatments of light predict shot noise as the irreducible limit for measurements of photon number or field intensity, stemming from the Poissonian photon statistics in coherent laser light, which approximates a classical field but retains this quantum fluctuation. However, in precision applications like optical interferometry, quantum effects further restrict sensitivity to the standard quantum limit imposed by shot noise, preventing improvements beyond certain thresholds without non-classical resources. Squeezing arises as a physical necessity to circumvent these limits by asymmetrically redistributing quantum uncertainties: reducing noise in one quadrature below the vacuum value (sub-shot-noise squeezing) at the expense of increased noise in the orthogonal quadrature, thus trading fluctuations to enhance measurement precision in the relevant observable. This enables performance beyond classical bounds in noise-sensitive systems.^[7] The squeeze operator emerged historically to describe non-classical light states extending beyond coherent states, which exhibit symmetric noise akin to classical waves but fail to access the full range of quantum correlations. Initially formulated as two-photon coherent states, this framework captured novel quantum optical effects, motivating the exploration of reduced-noise fields for advanced applications. The squeeze operator provides the unitary tool to impose this asymmetry in quadrature uncertainties.^[8]

Mathematical formulation

Operator definition

The squeeze operator \hat{S}(z) for a single bosonic mode is formally defined as

\hat{S}(z) = \exp\left( \frac{1}{2} z^* \hat{a}^2 - \frac{1}{2} z (\hat{a}^\dagger)^2 \right),

where \hat{a} and \hat{a}^\dagger are the annihilation and creation operators satisfying the canonical commutation relation [\hat{a}, \hat{a}^\dagger] = 1, and the complex parameter z = r e^{i\theta} encodes the squeezing properties with r \geq 0 determining the squeeze factor and \theta specifying the phase of the squeezed quadrature. This operator arises in the context of quadratic Hamiltonians in quantum optics and implements a Bogoliubov transformation on the bosonic ladder operators, which mixes \hat{a} and \hat{a}^\dagger to generate nonclassical correlations in bosonic systems. The generator \hat{K} = \frac{1}{2} \left( z^* \hat{a}^2 - z (\hat{a}^\dagger)^2 \right) is anti-Hermitian, satisfying \hat{K}^\dagger = -\hat{K}, which guarantees the unitarity of \hat{S}(z) since \hat{S}^\dagger(z) = \hat{S}(-z). When r = 0, z = 0, and \hat{S}(0) reduces to the identity operator. This operator plays a key role in transforming the ladder operators to yield squeezed vacuum states upon application to the bosonic vacuum.

Action on ladder operators

The action of the squeeze operator on the ladder operators of the quantum harmonic oscillator is derived using the Baker-Campbell-Hausdorff (BCH) formula, which expands the similarity transformation \hat{S}^\dagger(z) \hat{O} \hat{S}(z) for any operator \hat{O} in terms of nested commutators. The squeeze operator is defined as \hat{S}(z) = \exp\left( \frac{1}{2} (z^* \hat{a}^2 - z (\hat{a}^\dagger)^2 ) \right), where z = r e^{i\theta} parameterizes the squeezing strength r \geq 0 and phase \theta. Let A = \frac{1}{2} (z^* \hat{a}^2 - z (\hat{a}^\dagger)^2 ), so \hat{S}(z) = e^A and \hat{S}^\dagger(z) = e^{-A}. The BCH expansion for \hat{S}^\dagger(z) \hat{a} \hat{S}(z) = e^{\mathrm{ad}_{-A}} \hat{a} yields \hat{a} + [-A, \hat{a}] + \frac{1}{2!} [-A, [-A, \hat{a}]] + \cdots. The fundamental commutators are [A, \hat{a}] = z \hat{a}^\dagger and [A, \hat{a}^\dagger] = z^* \hat{a}, leading to nested terms that sum to hyperbolic functions: \hat{S}^\dagger(z) \hat{a} \hat{S}(z) = \hat{a} \cosh r + e^{i\theta} \hat{a}^\dagger \sinh r.^[9]^[8] By taking the Hermitian adjoint, the transformation of the creation operator follows immediately: \hat{S}^\dagger(z) \hat{a}^\dagger \hat{S}(z) = \hat{a}^\dagger \cosh r + e^{-i\theta} \hat{a} \sinh r. These Bogoliubov transformations preserve the canonical commutation relations [\hat{S}^\dagger(z) \hat{a} \hat{S}(z), \hat{S}^\dagger(z) \hat{a}^\dagger \hat{S}(z)] = 1, as required for bosonic operators, with the hyperbolic coefficients satisfying \cosh^2 r - \sinh^2 r = 1.^[8] The squeeze operator does not commute with the displacement operator \hat{D}(\alpha) = \exp(\alpha \hat{a}^\dagger - \alpha^* \hat{a}), which generates coherent states. Instead, they satisfy the braiding relation \hat{S}(z) \hat{D}(\alpha) = \hat{D}(\gamma) \hat{S}(z) e^{i\phi}, where \gamma = \alpha \cosh r + \alpha^* e^{i\theta} \sinh r and \phi = \frac{1}{2} (\alpha^* \gamma - \alpha \gamma^*) is a phase factor. This relation arises from combining the transformations of \hat{a} and \hat{a}^\dagger under conjugation by \hat{S}(z), and it facilitates the construction of displaced squeezed states.^[8]

Squeezed states

Vacuum squeezed states

The vacuum squeezed state, denoted as |\xi\rangle, is defined as the result of applying the squeeze operator \hat{S}(z) to the quantum vacuum state |0\rangle, where z = r e^{i\theta} with r the squeezing parameter and \theta the squeezing phase. This state represents a pure squeezed state with zero mean field amplitude, distinguishing it from displaced variants. In the Fock basis, the vacuum squeezed state exhibits an even photon number distribution, containing only components in even-number states |2n\rangle. The expansion coefficients are given by

\langle 2n | \xi \rangle = \frac{1}{\sqrt{\cosh r}} \frac{\sqrt{(2n)!}}{2^n n!} \left( -\frac{1}{2} e^{i\theta} \tanh r \right)^n,

with all odd-number coefficients vanishing. This structure arises from the pairing of creation and annihilation operators in the squeeze operator's disentangled form. For \theta = 0, the quadrature variances demonstrate the squeezing effect, with the amplitude quadrature \hat{X} = (\hat{a} + \hat{a}^\dagger)/2 reduced below the vacuum level and the phase quadrature \hat{P} = -i(\hat{a} - \hat{a}^\dagger)/2 correspondingly increased:

\langle (\Delta \hat{X})^2 \rangle = \frac{e^{-2r}}{4}, \quad \langle (\Delta \hat{P})^2 \rangle = \frac{e^{2r}}{4}.

These variances satisfy the Heisenberg uncertainty relation \langle (\Delta \hat{X})^2 \rangle \langle (\Delta \hat{P})^2 \rangle = 1/16, confirming minimum uncertainty while redistributing noise. The vacuum squeezed state displays non-classical features in its photon statistics, including a mean photon number \langle \hat{n} \rangle = \sinh^2 r, which is zero for r = 0 (reducing to the vacuum) but grows with squeezing. The photon number variance exceeds the mean, \Delta n^2 = 2 \sinh^2 r (\sinh^2 r + 1), leading to bunching effects characterized by a second-order correlation function g^{(2)}(0) > 1. These properties highlight the state's quantum correlations absent in classical light fields.

Coherent squeezed states

Coherent squeezed states, also known as squeezed coherent states, are quantum states of the harmonic oscillator that combine the effects of displacement and squeezing operations on the vacuum state. They are defined as |\alpha, \xi \rangle = \hat{D}(\alpha) \hat{S}(z) |0\rangle, where \hat{D}(\alpha) = \exp(\alpha \hat{a}^\dagger - \alpha^* \hat{a}) is the displacement operator that shifts the vacuum to a coherent state with amplitude \alpha, and \hat{S}(z) is the squeeze operator with complex parameter z = r e^{i \theta}, r being the squeeze parameter and \theta the squeeze angle. These states generalize the vacuum squeezed states, reducing to them when \alpha = 0.^[10] Due to the non-commutativity of the displacement and squeeze operators, the order of application affects the state properties. For the defined order \hat{D}(\alpha) \hat{S}(z) |0\rangle, the mean field is \langle \hat{a} \rangle = \alpha. The quadrature variances remain identical to those of the vacuum squeezed state, \Delta X_1^2 = \frac{1}{4} e^{-2r} and \Delta X_2^2 = \frac{1}{4} e^{2r} (for \theta = 0), but are centered around the displaced mean values determined by \alpha. For example, with \theta = 0, \langle X \rangle = \operatorname{Re}(\alpha) and \langle P \rangle = \operatorname{Im}(\alpha).^[10] A key property is the mean photon number, given by \langle \hat{n} \rangle = |\alpha|^2 + \sinh^2 r. This reflects the photon content from both displacement and squeezing without additional cross terms for this operator order.^[10] The expansion in the Fock basis is more intricate than for coherent states alone, involving associated Laguerre polynomials: the coefficients c_n = \langle n | \alpha, \xi \rangle are expressed using confluent hypergeometric functions or sums over Laguerre polynomials L_n^{(k)}(\cdot), which capture the non-classical photon statistics and sub-Poissonian behavior in certain regimes. These states are particularly useful in quantum optics for modeling laser fields with reduced noise in specific quadratures.^[10]

Applications

Quantum noise reduction

Squeezed states of light enable quantum noise reduction in optical systems by suppressing fluctuations in one field quadrature below the vacuum level, thereby improving signal-to-noise ratios in precision measurements such as interferometry. In these systems, vacuum fluctuations entering unused ports contribute to shot noise, which limits sensitivity; replacing ordinary vacuum with squeezed vacuum reduces this noise in the relevant quadrature without altering the Heisenberg uncertainty principle, as the orthogonal quadrature experiences increased fluctuations. In homodyne detection, squeezed light reduces phase noise in balanced detectors by injecting squeezed vacuum into the unused input port, minimizing quantum fluctuations that would otherwise add to the phase measurement uncertainty.^[11] This technique enhances the detection of weak signals, such as in quantum optics experiments, where balanced homodyne setups measure the quadrature noise directly, achieving observed reductions of up to 10 dB in the squeezed quadrature.^[11] Gravitational wave detectors like LIGO employ this approach by injecting squeezed vacuum into the unused antisymmetric port of the interferometer, improving strain sensitivity through the reduction of amplitude quadrature noise at frequencies relevant to astrophysical signals. In LIGO's implementation during the fourth observing run (O4, as of 2025), this has yielded a broadband quantum noise reduction of up to 6 dB in the audio band, extending the detector's effective range for binary neutron star mergers.^[12] More advanced frequency-dependent squeezing schemes further optimize this by varying the squeezed quadrature across frequencies, achieving up to 3 dB below the standard quantum limit (SQL) between 35 and 75 Hz while maintaining broadband performance.^[13] The theoretical limit of noise reduction is quantified by the squeezing parameter r, where the variance in the squeezed quadrature is V = \frac{1}{4} e^{-2r} (in units where vacuum variance is \frac{1}{4}), and the reduction in decibels is given by -10 \log_{10}(e^{-2r}). For example, 3 dB squeezing corresponds to r \approx 0.345, halving the uncertainty in one quadrature and thus the noise power relative to vacuum.^[13] Compared to classical limits, squeezed states surpass the SQL in phase estimation, which for interferometers bounds precision to \Delta \phi \geq 1 / \sqrt{N} (where N is the mean photon number); squeezing enables \Delta \phi < 1 / \sqrt{N}, approaching the Heisenberg limit \Delta \phi \geq 1 / N for enhanced metrology. This quantum advantage has been demonstrated in interferometric setups, where injected squeezed vacuum reduces phase noise below the SQL by factors tied to the squeezing level.^[14]

Quantum metrology and sensing

In quantum metrology, squeezed states produced via the squeeze operator enhance the precision of parameter estimation by reducing quantum noise in the relevant quadrature, surpassing the standard quantum limit (SQL) imposed by uncorrelated resources. For example, in optical interferometry, the injection of squeezed vacuum into the dark port of a Mach-Zehnder interferometer minimizes phase noise, enabling sub-shot-noise sensitivity for estimating phase shifts.^[7] In atomic ensembles, spin squeezing—generated by nonlinear interactions analogous to the squeeze operator acting on collective spin operators—improves phase estimation in Ramsey interferometry, where the spin-squeezing parameter \xi^2 < 1 quantifies the metrological gain for detecting relative phase shifts between hyperfine states.^[15] A key advantage of squeezed states in metrology is their ability to achieve Heisenberg-limited scaling, where the precision \Delta \theta for estimating a parameter \theta improves as $1/N using N probes, compared to the SQL scaling of $1/\sqrt{N}. This enhancement stems from the non-classical correlations introduced by squeezing, which redistribute uncertainty to optimize the Cramér-Rao bound in quantum Fisher information-based protocols. Squeezed states find direct applications in quantum sensing for detecting weak signals. In magnetometry, polarization-squeezed light probing an unpolarized rubidium vapor via the Faraday effect has demonstrated a 3.2 dB improvement in sensitivity for measuring weak radiofrequency magnetic fields, limited primarily by atomic noise rather than shot noise.^[16] Similarly, in optomechanical systems, injecting squeezed light into a cavity reduces backaction noise on the mechanical resonator, enhancing force sensitivity; for instance, schemes using frequency-dependent squeezing have enabled sub-SQL detection of weak forces, with theoretical sensitivities reaching the zeptonewton level by correlating optical and mechanical quadratures.^[17]^[18] For distributed or multipartite sensing, the two-mode squeeze operator \hat{S}_2(\zeta) = \exp(\zeta^* \hat{a}_1 \hat{a}_2 - \zeta \hat{a}_1^\dagger \hat{a}_2^\dagger) generates bipartite entangled states from the vacuum, fostering EPR-like correlations that enable Heisenberg-limited precision in estimating parameters like phase differences across modes. These states are particularly valuable in protocols for simultaneous estimation of multiple parameters or in noisy environments, where entanglement monogamy is mitigated by the Gaussian nature of the squeezing.^[19]

History and experiments

Theoretical origins

The theoretical origins of the squeeze operator lie in the foundational developments of quantum optics during the 1920s, particularly Paul Dirac's quantization of the electromagnetic field. In his 1927 paper, Dirac formulated the quantum theory of radiation emission and absorption, introducing photons as quanta of the field and establishing the commutation relations for creation and annihilation operators that underpin later quantum optical formalisms.^[20] Mathematical groundwork for the squeeze operator advanced in the 1960s with Robert M. Wilcox's analysis of exponential operators in quantum physics. Wilcox's 1967 work derived parameter differentiation techniques for unitary operators of the form \exp(t G), where G is a Lie algebra generator, providing the disentangling formulas essential for expressing the squeeze operator in normal-ordered form. The concept of squeezing for optical fields was formalized in the 1970s through studies of minimum-uncertainty states with unequal quadrature variances. Daniel Stoler introduced equivalence classes of such states in 1970, while E. Y. C. Lu extended the analysis in 1971; these works highlighted nonlinear transformations reducing noise in one field quadrature.^[21] Concurrently, Daniel F. Walls and collaborators explored squeezing in nonlinear optical processes, and H. P. Yuen defined two-photon coherent states in 1976, which are equivalent to squeezed vacuum states and demonstrated reduced uncertainty below the coherent-state limit.^[8] The squeeze operator evolved from Bogoliubov transformations, initially developed by N. N. Bogoliubov in 1958 for superconductivity to diagonalize fermionic Hamiltonians via quasiparticle pairings. In quantum optics, these transformations were adapted to bosonic modes, describing parametric down-conversion where the squeeze operator mixes creation and annihilation operators to generate correlated photon pairs. Seminal reviews, such as the 2005 textbook by Christopher C. Gerry and Peter L. Knight, synthesize this framework, while Michael M. Nieto and D. Rodney Truax's 1997 overview integrates squeezing within generalized coherent states.

Experimental realizations

The first experimental realization of squeezed states was achieved in 1985 by Slusher et al., who generated squeezed light through nondegenerate four-wave mixing in a sodium atomic vapor within an optical cavity, observing 0.3 dB of quadrature squeezing below the vacuum noise level at frequencies around 40 MHz.^[22] This landmark demonstration confirmed the theoretical predictions of the squeeze operator by producing nonclassical light with reduced uncertainty in one quadrature. Subsequent refinements in optical parametric oscillators (OPOs) during the late 1980s and early 1990s enabled higher squeezing levels; for instance, Wu et al. reported approximately 0.6 dB of squeezing using a degenerate subthreshold OPO in 1987.^[23] In the 1990s, advances in cavity-enhanced OPOs significantly improved squeezing performance. Breitenbach et al. achieved 6 dB of vacuum squeezing at 860 nm using a monolithic standing-wave OPO with a nonlinear potassium titanyl phosphate crystal in 1995, operating below threshold to generate broadband squeezed vacuum.^[24] Building on this, Lam et al. demonstrated over 7 dB of squeezing at sideband frequencies near DC using a traveling-wave degenerate OPO with a potassium titanyl phosphate crystal in 1999, highlighting the potential for low-frequency applications in precision measurements. These cavity OPO systems became the standard for generating high-fidelity squeezed states due to their efficient nonlinear interaction and phase-matching capabilities. The 2010s saw the emergence of integrated photonics for compact, chip-scale squeezing. In 2015, Dutt et al. reported the first on-chip generation of squeezed light using a silicon nitride microring resonator as a four-wave mixing OPO, measuring 1.7 dB of intensity-difference squeezing (corrected to 5 dB accounting for losses) between twin beams at telecommunications wavelengths.^[25] This platform leveraged high-quality-factor resonators for enhanced nonlinearity in a monolithic CMOS-compatible design, paving the way for scalable quantum photonic devices. Further progress included broadband quadrature squeezing up to 3 dB in silicon nitride microrings by Lu et al. in 2020, demonstrating sub-shot-noise correlations over gigahertz bandwidths.^[26] Verification of squeezed states relies on techniques that probe quantum correlations beyond classical limits. Homodyne tomography, introduced by Smithey et al. in 1993, measures the Wigner function of squeezed vacuum states by balanced homodyne detection at multiple phases, revealing negativities that confirm nonclassicality; for example, they reconstructed the Wigner function for up to 1.5 dB squeezed states from an OPO.^[27] For number-squeezed states, photon counting detects sub-Poissonian statistics, where the Mandel Q parameter is negative; this was demonstrated in early OPO experiments, such as those by Wu et al., showing variance reductions below the coherent state limit through direct photodetection of photon number fluctuations.^[23] Recent developments up to 2025 have pushed squeezing levels beyond 15 dB in continuous-wave systems tailored for gravitational-wave detectors like LIGO upgrades. Vahlbruch et al. generated 15 dB of quadrature squeezing at 1064 nm using a bow-tie OPO with a periodically poled potassium titanyl phosphate crystal in 2016, achieving this in the megahertz frequency range suitable for interferometer injection.^[28] For LIGO's frequency-dependent squeezing enhancements, sources deliver over 15 dB internally, with effective noise reductions up to 6 dB observed in detectors after losses, as implemented in the 2023 observing run (O4); further upgrades in 2024 achieved up to 3 dB below the standard quantum limit in the 35-75 Hz band using advanced frequency-dependent squeezing.^[14]^[13] In parallel, atomic ensembles have enabled squeezing via Rydberg interactions; Jenkins et al. reported spin squeezing in cesium atom arrays through Rydberg dressing in 2023, achieving a metrological gain equivalent to 4 dB reduction in phase uncertainty using van der Waals blockade for collective nonlinearities.^[29] Additionally, in 2025, an ultra-stable squeezed vacuum source using ^{87}Rb atomic vapor at 795 nm demonstrated 12.5 dB squeezing with high phase stability, advancing quantum sensing applications.^[30]

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